A370259 a(n) = (T(n,n+1) - 1)/n^3 for n >= 1, where T(n,x) is the n-th Chebyshev polynomial of the first kind.

1, 2, 9, 75, 961, 16900, 380689, 10498005, 343323841, 13007560326, 560789801881, 27125634729375, 1455389462287489, 85805768251305992, 5515372218107327521, 383931652351786775721, 28778117694539885440129, 2311202255914842794592010, 198009919900727928789497641, 18027589454633803742596931571

Offset: 1

Author

Peter Bala, Mar 11 2024

Keywords

• nonn
• easy

Comments

It appears that a(2*n+1) is always a square, while a(2*n) = (n + 1) * a square. See A370260 and A370261.

Links

• Paolo Xausa, Table of n, a(n) for n = 1..350

Crossrefs

Cf. A008310, A342205, A370260, A370261.

Programs

Maple

seq( simplify( (ChebyshevT(n, n+1) - 1)/n^3 ), n = 1..20);

Mathematica

Array[(ChebyshevT[#, #+1]-1)/#^3 &, 20] (* Paolo Xausa, Mar 14 2024 *)

Python

from sympy import chebyshevt
def A370259(n): return (chebyshevt(n,n+1)-1)//n**3 # Chai Wah Wu, Mar 13 2024

Formula

a(n) = Sum_{k = 1..n} (2^k)*n^(k-2)*binomial(n+k, 2*k)/(n + k) (shows that a(n) is an integer).

a(n) = (cos(n*arccos(n+1)) - 1)/n^3.

a(n) = (A342205(n) - 1)/n^3.

a(n) = ( (n + 1 + sqrt(n*(n+2)))^n + (n + 1 - sqrt(n*(n+2)))^n - 2 )/(2*n^3).